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<h1 class="reftitle">distance</h1>
<h2>Purpose</h2>
<p>Computes the closest distance between the convex set and given point.</p>
<h2>Syntax</h2>
<pre class="synopsis">ret = distance(Set,x)</pre>
<pre class="synopsis">ret = Set.distance(x)</pre>
<h2>Description</h2>
<p></p>
        Compute the closest distance between the convex <tt>Set</tt> and given point <tt>x</tt>
        The approach is based on solving the optimization problem
        <p class="programlistingindent"><img src="../../../../../../fig/mpt/modules/geometry/sets/@ConvexSet/distance5.png" alt="../../../../../../fig/mpt/modules/geometry/sets/@ConvexSet/distance5.png"></p>
        where <img src="../../../../../../fig/mpt/modules/geometry/sets/@ConvexSet/distance1.png" alt="../../../../../../fig/mpt/modules/geometry/sets/@ConvexSet/distance1.png"> is the given point in the same dimension as the set and <img src="../../../../../../fig/mpt/modules/geometry/sets/@ConvexSet/distance2.png" alt="../../../../../../fig/mpt/modules/geometry/sets/@ConvexSet/distance2.png"> is the 
        point inside the <tt>Set</tt>. If the optimization terminated successfully, the output 
        contains the distance <tt>ret.d</tt> and the points <tt>ret.y</tt>, <tt>ret.x</tt>.
        Otherwise, the output is empty. 
        If the <tt>Set</tt> is an array of convex sets, the distance and the point <tt>y</tt> are
        returned in a cell arrays.
	<h2>Input Arguments</h2>
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<td><tt>Set</tt></td>
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<p></p>Any object derived from the <tt>ConvexSet</tt> class, e.g. <tt>Polyhedron</tt>, <tt>YSet</tt>, ...<p>
	    		Class: <tt>ConvexSet</tt></p>
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<td><tt>x</tt></td>
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<p></p>A point given as a real vector with the same dimension as the convex set.<p>
	    		Class: <tt>double</tt></p>
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<h2>Output Arguments</h2>
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<td><tt>ret</tt></td>
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<p></p>Structure that contains the information about the computed distance and the points <img src="../../../../../../fig/mpt/modules/geometry/sets/@ConvexSet/distance3.png" alt="../../../../../../fig/mpt/modules/geometry/sets/@ConvexSet/distance3.png">, <img src="../../../../../../fig/mpt/modules/geometry/sets/@ConvexSet/distance4.png" alt="../../../../../../fig/mpt/modules/geometry/sets/@ConvexSet/distance4.png">.<p>
	    		Class: <tt>struct</tt><p></p><tr valign="top">
<td><tt>ret.exitflag</tt></td>
<td>
<p></p>Termination status from the related optimization problem.<p>
	    		Class: <tt>double</tt></p>
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<td><tt>ret.dist</tt></td>
<td>
<p></p>Distance between the point <tt>x</tt> and the convex <tt>Set</tt>.<p>
	    		Class: <tt>double</tt></p>
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<td><tt>ret.y</tt></td>
<td>
<p></p>The point that is contained in the convex <tt>Set</tt> and is closest to <tt>x</tt>.<p>
	    		Class: <tt>double</tt></p>
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<td><tt>ret.x</tt></td>
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<p></p>The point <tt>x</tt> that was provided.<p>
	    		Class: <tt>double</tt></p>
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<h2>Example(s)</h2>
<h3>Example 
				1</h3> Construct a convex set by intersecting a circle and random linear inequalities.  Define variable first<pre class="programlisting"> z = sdpvar(2,1); </pre>
<pre class="programlisting"></pre> Define a set <tt>S</tt> using <tt>YSet</tt> class  <pre class="programlisting"> options = sdpsettings('solver','sedumi','verbose',0); </pre>
<pre class="programlisting"></pre>
<pre class="programlisting"> S = YSet(z, [norm(z)&lt;=1; randn(2)*z&lt;=0.5*ones(2,1)], options); </pre>
<pre class="programlisting"></pre> Compute the distance to a point [-5;8] <pre class="programlisting"> x = [-5; 8];</pre>
<pre class="programlisting"></pre>
<pre class="programlisting"> d = S.distance(x)</pre>
<pre class="programlisting">
d = 

    exitflag: 1
        dist: 8.43398113093422
           x: [2x1 double]
           y: [2x1 double]

</pre> We can plot the set and the points <tt>x</tt>, <tt>y</tt> 
      <pre class="programlisting"> S.plot('alpha',0.5,'color','green'); 
              hold on;  text(x(1),x(2),'x');
              axis([-3 2 -2 3]); text(d.y(1),d.y(2),'x');
        </pre>
<pre class="programlisting">Plotting...
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<p class="programlistingindent"><img src="../../../../../../fig/mpt/modules/geometry/sets/@ConvexSet/distance_img_1.png" alt="../../../../../../fig/mpt/modules/geometry/sets/@ConvexSet/distance_img_1.png" width="60%"></p>
<h2>See Also</h2>
<a href="./outerapprox.html">outerapprox</a>, <a href="./support.html">support</a>, <a href="./separate.html">separate</a><p></p>
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<br><p>©  <b>2010-2013</b>     Colin Neil Jones: EPF Lausanne,    <a href="mailto:colin.jones@epfl.ch">colin.jones@epfl.ch</a></p>
<p>©  <b>2010-2013</b>     Martin Herceg: ETH Zurich,    <a href="mailto:herceg@control.ee.ethz.ch">herceg@control.ee.ethz.ch</a></p>
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